When do workers versus capital owners share of income increase?

Question

The share of total income obtained by workers rather than capital owners is for obvious reasons of interest. Assuming that the economy can be desribed by an aggregate production function

$$Y = F(K,L)$$

where $K$ is capital input and $L$ is labor input the total expenditure on labor and capital is given as

$$pY = rK + wL,$$

where $p$ is output price, $r$ is user cost of capital and $w$ is the wage (perfect competition is assumed on all markets). Labors share of total revenue or expenditure is therefore given as

$$s_L := \frac{wL}{pY} = \frac{wL}{rK + wL},$$

depending only on $w,r,L,K$.

The elasticity of substitution is defined as

$$\sigma := - \frac{d\log(K/L)}{d\log(r/w)},$$

describing how the capital to labor ratio changes given a change in the ratio of user cost of capital to wage.

Since both labor share of expenditure and elasticity of substitution depend on the variables $w,r,L,K$ this suggest the question:

How does changes in labors share of income $s_L$ as a result of changes in relative price $(r/w)$ relate to the elasticity of substitution $\sigma$?

Answer 1

From the shares equation, we obtain: $$ 1 + \frac{rK}{wL} = \frac{1}{s_L} \to tk = \frac{1 - s_L}{s_L} $$ where I defined $t = \frac{r}{w}$ and $k = \frac{K}{L}$.

then taking logs gives: $$ \ln(k) = -\ln(t) + \ln(1-s_L) - \ln(s_L) $$ Then take the derivative of this expression with respect to $\ln(t)$: $$ \begin{align*} -\sigma &= -1 - \dfrac{\dfrac{\partial s_L}{\partial \ln(t)}}{1 - s_L} - \gamma,\\ &=-1 - \gamma - \frac{s_L}{1 - s_L} \gamma = -\left(\frac{\gamma}{1 - s_L} +1\right) \end{align*} $$ Where we defined $\gamma$ to be the elasticity of the labour revenue share with respect to the $r/w$ ratio.

Solving for $\gamma$ gives: $$ \gamma = \left(\sigma-1\right)(1 - s_L) $$ which means that $\gamma > 0$ if $\sigma < 1$. In other words, if the elasticity of subsititution is smaller than one, then the labour share increases when $r/w$ increases.